Do Limits to Arbitrage Explain the Benefits of Volatility-Managed Portfolios?

Transcription

1 Do Limits to Arbitrage Explain the Benefits of Volatility-Managed Portfolios? Pedro Barroso University of New South Wales Andrew Detzel University of Denver November 22, 2017 Abstract Rational asset pricing models do not explain the alphas and utility gains produced by volatility managing stock portfolios found by prior studies. Hence, we hypothesize that limits to arbitrage (LTA) explain this abnormal performance. Contrary to the hypothesis, we document that volatility managing long-equity portfolios only improves performance among stocks with low and medium LTA proxied by institutional ownership (IO) and idiosyncratic volatility (IV). Moreover, the utility gains of volatility managing mean-variance-efficient portfolios of these factors out-of-sample are consistently twice to many times as high for factors constructed from low-lta stocks than high-lta stocks. The concentration of the economic gains from volatility management in market segments with the lowest LTA contrasts sharply with the common finding that anomaly returns are lowest in these segments. JEL classification: G11, G12, G14 Keywords: Volatility-managed portfolios, limits to arbitrage, anomalies and (corresponding author), respectively. For helpful comments and suggestions, we thank: Paul Karehnke, Alan Moreira, and Tyler Muir.

2 1. Introduction Several recent studies find that volatility managing equity portfolios taking more risk when volatility is low, and vice versa produces significant alphas and large increases in investor utility (e.g., Fleming et al., 2001, 2003; Kirby and Ostdiek, 2012; Barroso and Santa-Clara, 2015; Moreira and Muir, 2017a,b; Barroso and Maio, 2017a,b). This result obtains for the market portfolio, the Fama and French (1993, 2015) five factors, the Hou et al. (2015) four factors, and the betting-against-beta factor of Frazzini and Pedersen (2014), among others. These volatility-management benefits occur because volatility is persistent from month to month but only weakly related to expected future returns. This stylized fact implies that the price of risk falls in times of high volatility, contrary to extant (frictionless) rational models of asset prices. 1 Violations of rational models are not surprising, however, if limits to arbitrage (LTA) prevent traders from correcting pricing anomalies (e.g., Shleifer and Vishny, 1997). For example, limits to short sales, which are well known to allow over-pricing to persist, could prevent prices from falling sufficiently to make risk premia rise with volatility to the extent predicted by rational models (e.g., Miller, 1977; Nagel, 2005). In this paper, we test the hypothesis that LTA cause the benefits of volatility management. We proxy for LTA with idiosyncratic volatility (IV) and institutional ownership (IO). Idiosyncratic volatility limits arbitrage because it is a risk to under-diversified traders attempting to exploit mispricing (e.g., Shleifer and Vishny, 1997; Stambaugh et al., 2015). IV also offers the benefit of availability over the entire CRSP sample (1926 present). IO limits arbitrage, especially for overpriced stocks, because it is a crucial part of the supply of loanable shares in short-sales (e.g., Nagel, 2005). Consistent with these LTA, many studies show that mispricing proxied by anomaly returns increases with IV and decreases with IO. 2 Our main strategy for testing our hypothesis is to sort stocks into low-, medium-, and high-lta groups, and then compare differences in performance gains from volatility management by group. We examine the performance of volatility-managed portfolios following the methodology of 1 Moreira and Muir (2017b) show that that the following rational models of asset prices predict a weakly positive risk-return tradefoff: The habits model (Campbell and Cochrane, 1999), the long-run-risk model (Bansal et al., 2012), the time-varying rare disasters model (Wachter, 2013), and the intermediary asset-pricing model (He and Krishnamurthy, 2013). 2 These studies are partially reviewed in the next section. 1

3 Moreira and Muir (2017b). We scale portfolios proportionally to the inverse of their variance to generate volatility-managed portfolios. We then regress excess returns on the managed portfolios on those of their unmanaged counterparts. A positive alpha in these regressions indicates that volatility management expands the mean-variance frontier for investors relative to the unmanaged portfolios. However, one can not assess the relative economic significance of different alphas by directly comparing them across regressions. Exploiting abnormal returns requires bearing residual risk that differs by asset. Thus, to facilitate cross-asset comparison of alphas, we measure the economic significance of volatility management with appraisal ratios and utility gains of meanvariance investors, which both account for residual risk. Also following Moreira and Muir (2017b), we present key results using IV for the 90-year sample and each non-overlapping 30-year subsample to examine robustness and time-variation. We contribute four main findings to the literature, which collectively reject our hypothesis. First, we examine volatility management of long positions in IV- and IO-tercile portfolios. Long strategies are particularly interesting because they are easier and less costly to implement than short strategies. Moreover, most investors only take long positions (e.g., Barber and Odean, 2008; Stambaugh et al., 2012). Contrary to our hypothesis, we find that the volatility-managed high-iv portfolio earns economically and statistically insignificant alpha relative to its unmanaged counterpart over and each subsample. The same result obtains for the volatility-managed low-io portfolio over In contrast, the low- and medium-lta volatility-managed portfolios earn significant alphas over and each subsample except The (meanvariance) utility gains for the volatility-managed low- and medium-lta portfolios range from 46% to 175%. For comparison, Campbell and Thompson (2008) find that the utility gains of timing the aggregate stock market are about 30%. Second, we find that volatility managing low-minus-high (high-minus-low) IV (IO) factors generates significant alphas over ( ) and each subsample we consider, extending similar prior results for other factors. Next, we investigate our hypothesis applied to volatility management of long-short anomaly factors. We double-sort stocks into 3x5 value-weighted portfolios by first sorting into IV or IO terciles 3 Moreira and Muir (2017b) also find that volatility management generally yields insignificant gains in the sample due to low variation in volatility over that time. 2

4 and then independently into quintiles based on size, book-to-market, momentum return, profitability, and investment. We first document that these sorts exhibit evidence of the effects of LTA; the spreads in CAPM alphas produced by the characteristic sorts increase with IV and decrease with IO. Our third main finding is that no consistent relationship exists between either IV or IO and the volatility-management benefits of long-short anomaly strategies. Moreover, conditioning on IV or IO frequently results in insignificant alphas of volatility-managed factors. The anomaly factors in isolation are not inherently useful to investors who are ultimately concerned about the mean-variance efficiency of their entire portfolios. Hence, we assess how volatility management improves performance of mean-variance efficient (MVE) portfolios constructed from anomaly factors within each IV or IO tercile. Our fourth main finding, described further below, is that the economic significance of volatility managing MVE portfolios is largely concentrated in low- and medium-lta stocks. We use two sets of factors in constructing MVE portfolios. The first set, denoted FF3+MOM, consists of the IV- or IO-tercile itself as well as the long-short size, book-to-market, and momentum portfolios for the tercile. The second set of portfolios, denoted FF5+MOM, adds the corresponding long-short profitability and investment portfolios. For each IV tercile, volatility managing in-sample MVE portfolios produces statistically significant alphas. However, the associated utility gains are highest for the low-iv portfolios (55% vs 43% for the FF3+MOM factors and 26% vs 9% for the FF5+MOM factors). Both the high- and low-io managed MVE portfolios also earn significant alphas. However, the utility gains of volatility managing the high-io MVE portfolios are two to three times as high as those of managing the low-io portfolios (59% vs 18% for the FF3+MOM factors and 63% vs 31% for the FF5+MOM factors). As noted by Moreira and Muir (2017b), the performance of unmanaged in-sample MVE portfolios is much higher than what would be attainable in real-time, likely biasing downward the gains of volatility management. Hence, each month, we generate recursively estimated out-of-sample (OOS) MVE portfolios using the same factors as the in-sample analysis. We also construct equal-weighted 1/N portfolios, which DeMiguel et al. (2009) show frequently outperform MVE portfolios out of sample. In order to use the maximum sample and avoid a profusion of Tables, in the main 3

5 body of the paper we only discuss OOS-MVE- and 1/N-portfolio results that use the FF3+MOM factors. We relegate discussion of the associated results using the FF5+MOM factors, which are qualitatively similar, to the Appendix. Over the sample, for each IV tercile, all three volatility-managed OOS-MVE portfolios earn significant alpha. However, the high-iv managed 1/N portfolio earns insignificant alpha and the high-iv volatility-managed OOS-MVE and 1/N portfolios actually have lower Sharpe ratios than their unmanaged counterparts (1.15 to 1.10 and 1.02 to 0.78, respectively). Moreover, the utility gains from volatility management are much higher for the low-iv OOS-MVE and 1/N portfolios than the corresponding high-iv portfolios (65% vs 19% and 49% vs 2%, respectively). Examining subsamples shows that the benefits of volatility managing OOS-MVE portfolios increase over time. Over the subsample, the volatility-managed low-iv OOS-MVE and 1/N portfolios earn significant alpha and massive utility gains of 198% and 200%, respectively. In contrast, during the same period, volatility managing the high-iv OOS-MVE and 1/N portfolios yields insignificant alpha and effectively no increase in utility. The performance gains of volatility managing OOS-MVE and 1/N portfolios vary with IO in a consistent manner as with IV. For each IO tercile, the volatility-managed OOS-MVE portfolios earn statistically significant alphas relative to the unmanaged portfolios. However, volatility management yields larger utility gains in the high-io OOS-MVE and 1/N portfolios than for the corresponding low-io portfolios (233% vs 64% and 45% vs 27%, respectively). For both IV and IO, the volatility-managed low-lta OOS-MVE and 1/N portfolios earn economically larger utility gains than the corresponding high-lta portfolios. Moreover, by the nature of LTA, the gains of high-lta stocks are also unlikely to be realizable. IV is highly correlated with transaction costs, which would only increase the difference in benefits from volatility management between low- and high-iv portfolios. 4 MVE portfolios are also constructed from anomalies whose performance critically depends on the success of short positions. In low-io stocks, investors would find it very expensive, if not impossible, to actually execute the necessary short sales to obtain any documented benefits from volatility management. Thus our results understate the difference 4 Novy-Marx and Velikov (2016) show that IV explains 55% of the cross-sectional variation in transaction costs. 4

6 in economic significance between volatility managing high-io and low-io portfolios. Overall, the economic gains generated by volatility management are largely concentrated in low-lta stocks. Our study is related to the voluminous literature, which is partially reviewed below, that studies the effect that frictions have on anomalies. These studies typically find that anomaly returns increase with LTA. Our findings are unique in this literature because we find the anomalous returns of volatility-managed portfolios are concentrated in stocks with the lowest LTA that should have the least mispricing. The remainder of this paper proceeds as follows. Section 2 discusses our main hypothesis. Section 3 describes our data. Section 4 presents our main results. Section 5 concludes. 2. Limits to arbitrage and mispricing To paraphrase Baker and Wurgler (2006): mispricing requires an irrational or uninformed demand shock and a limit to arbitrage. Moreira and Muir (2017b) show that the benefits of volatility managing portfolios can not be explained by any of the leading rational asset pricing models and therefore likely represent mispricing. Thus, our main hypothesis is that: Hypothesis 1. The alphas, Sharpe-ratio increases, and utility gains from volatility managing equity portfolios increase with limits to arbitrage. This hypothesis directly relates to the vast literature that finds proxies for limits to arbitrage (LTA) such as idiosyncratic volatility, institutional ownership, transaction costs, and even size prevent the impacts of irrational demand on prices from being corrected. The theoretical strand of this literature argues that idiosyncratic volatility (IV) is a risk to traders attempting to exploit mispricing (e.g., DeLong et al., 1990; Pontiff, 1996; Shleifer and Vishny, 1997; Pontiff, 2006). Many empirical studies find evidence consistent with this LTA interpretation of IV. Specifically, anomaly returns tend to increase with IV (e.g., Pontiff, 1996; Wurgler and Zhuravskaya, 2002; Ali et al., 2003; Mashruwala et al., 2006; Zhang, 2006; Scruggs, 2007; McLean, 2010; Li and Zhang, 2010; Stambaugh et al., 2015; Larrain and Varas, 2013; Stambaugh and Yuan, 2017). Institutional ownership (IO) is also widely used as an (inverse) proxy for LTA, especially short- 5

7 sale constraints, because it is a crucial source of loanable shares (e.g., D Avolio, 2002; Asquith et al., 2005; Nagel, 2005; Duan et al., 2010; Hirshleifer et al., 2011). Short-sale constraints are particularly interesting in our setting for two reasons. First, anomaly returns are largely driven by their short legs and tend to be higher in the presence of short-sale constraints (e.g., Diether et al., 2002; Geczy et al., 2002; Ofek et al., 2004; Nagel, 2005; Avramov et al., 2013; Drechsler and Drechsler, 2017). Miller (1977), Chen et al. (2002), Duffie et al. (2002), and Scheinkman and Xiong (2003) justify these findings theoretically by showing that short-sale constraints prevent value-reducing information from being impounded in prices. Second, the only hypothesis that Moreira and Muir (2017b) do not at least partially rule out that explains the benefits of volatility management is slow trading. If volatility rises, but investors are too slow to sell, then prices will not fall enough to capture the full risk-premium predicted by rational models. Short-sale constraints are an important cause of slow selling. It is important to note that none of the literature on LTA and asset prices identifies anomalies that earn higher returns when LTA proxied by IV and IO are low. 3. Data We obtain daily and monthly data on individual common stocks from CRSP, annual accounting data from COMPUSTAT, and both daily and monthly returns on the Fama and French (1993, 2015) and Carhart (1997) factors (MKT, SMB, HML, MOM, CMA, and RMW ) as well as the risk-free rate (r f ) from the website of Kenneth French. We correct stock returns for delisting bias following Shumway (1997). We define momentum return (r 12,2 ), market cap (ME), bookto-market ratio (BM), operating profit (OP ), and investment (INV ) following Fama and French (2015, 2016). We measure idiosyncratic volatility for stock i in month t as the standard deviation (ɛ id ) of the residuals from a CAPM regression estimated using daily data (17 days minimum) in month t 1: 5 r id r fd = a it + β it MKT d + ɛ id, d t 1. (1) 5 Some studies define IV relative to a multi-factor model, such as the Fama-French 3-factor model (e.g., Ang et al., 2006). However, this practice potentially alters the interpretation of IV and thus studies that focus on the friction-aspect of IV often only use the market return as a factor (e.g. Novy-Marx and Velikov, 2016) 6

8 Institutional ownership (IO) is the percentage of shares owned by institutional owners and comes from Thomson Financial 13(f) Institutional Holdings at the quarterly frequency. Following Moreira and Muir (2017b), our maximum sample period, which uses IV, is 1926:1-2015:12. We also consider three 30-year subsamples: 1926:1 1955:12, 1956:1 1985:12, and 1986:1 2015:12. IO is only available for the most recent subsample. The factors CM A and RM W are only available for the period 1963:7 2015:12, while the other Fama-French factors are available over effectively the maximum sample (M OM is only available since 1927:1). 4. Main results 4.1. Portfolio construction Our volatility-management and empirical methodologies closely follow those of Moreira and Muir (2017b). We construct volatility-managed portfolios by scaling their excess returns by the inverse of their variance. Letting f t denote a buy-and-hold excess return in month t, the the managed portfolio return (f t ) is defined as: ft = c ˆ t 1 2 f t, (2) where ˆ t 1 denotes the volatility of daily returns over month t 1 and the constant c is chosen to equate the unconditional volatilities of f t and f t. The motivation for this strategy comes from optimal portfolio choice of a mean-variance investor. If f t is the market return, or uncorrelated with other factors, then the optimal weight in f t+1 is proportional to 1 γ t E t(f t+1 ) 2 t (f t+1), where γ t denotes relative risk aversion and E t (f t+1 ) ( t (f t+1 )) denotes conditional expectation (variance) of f t+1. Since expected returns are highly unpredictable at the monthly frequency and volatility is highly persistent, c ˆ 2 t 1 approximates the role of Et(f t+1) 2 t (f t+1) in Eq. (2) Empirical Methodology We regress the excess returns of volatility-managed portfolios on their unmanaged counterparts: f t = α + β f t + ɛ t. (3) 7

9 A positive alpha indicates that access to f t to that of a buy-and-hold position in f t. increases the maximum possible Sharpe ratio relative When f t is a systematic factor, such as the market portfolio, that summarizes common variation for many assets, a positive alpha implies that volatility management improves the mean-variance frontier. The ultimate benefit of volatility management to an investor is increased utility from a higher maximum Sharpe ratio for their whole portfolio. Thus, alpha only matters to the extent that it expands the mean-variance frontier. Intuitively, this expansion depends on the alpha relative to the residual risk investors must bear to capture it. The maximum Sharpe ratio (SR New ) attainable from access to f t and f t is given by: SR New = ( ) α 2 + SR 2 (ɛ t ) Old, (4) where SR Old is the Sharpe ratio of f t (e.g., Bodie et al., 2014). Hence, we use the appraisal ratio ) as one measure of volatility-management benefits to compare across assets. ( α (ɛ t) A disadvantage of the appraisal ratio is that its effect on Sharpe ratios is nonlinear. The same appraisal ratio has a greater impact on a lesser SR Old than vice versa. Thus, to further facilitate comparison across assets, we measure the percentage increase in mean-variance utility, which for any level of risk aversion is equal to: Utility gain = SR2 New SR2 Old SROld 2. (5) Campbell and Thompson (2008) find that timing expected returns on the stock market increases mean-variance utility by approximately 35%, providing a useful benchmark utility gain Long-equity portfolios We begin our analysis be assessing the benefits of volatility managing long portfolios based on IV and IO. Long portfolios are the basic building block of more complicated strategies and most interesting to the outstanding majority of investors who only take long positions. Moreover, the performance of long-only strategies does not require potentially costly or difficult short positions. 8

10 Each month, we sort every stock in CRSP into value-weighted IV or IO terciles, denoted, respectively, IV 1, IV 2, and IV 3 or IO 1, IO 2, and IO 3. Table 1 presents average excess returns and estimates of CAPM regressions for the unmanaged IV - and IO-tercile portfolios. Panel A shows that consistent with prior evidence (e.g., Ang et al., 2006), IV has a significant impact on returns. Over , average returns decrease with IV, and are even insignificant for IV 3. Similarly, Panel B shows that IV 1 earns a significant positive CAPM alpha of 1.23% per year, while this figure decreases to 7.56% per yearn for IV 3. Panel C shows analogous findings as Panel A, but for the IO terciles. Although insignificantly, IO 1 actually under-performs the risk-free rate over Average returns increase with IO with a significant spread of about 10.0% per year between IO 1 and IO 3. Panel D shows a parallel pattern in CAPM alphas, which significantly increase by 9.4% from -9.4% per year for IO 1 to an insignificant 0.0% per year for IO 3. The abysmally poor returns and negative alpha s of low-io stocks are consistent with limits to short selling. Figure 1 plots the cumulative log value of $1 invested at the beginning of the sample in each of the volatility-managed IV and IO portfolios relative to their unmanaged counterparts. Panel A shows that IV 1 and IV 2 steadily outperform the remaining portfolios over the 90-year window and each accumulate to about 10.7 log dollars ($44,356). The unmanaged IV 1 and IV 2 accumulate to about 8.8 log dollars ($6,634) ( 15% of $44,356). In contrast, IV 3 and IV 3 have much lower cumulative returns of about 1.9 log dollars ($7). The findings are similar in Panel B for IO portfolios. The volatility-managed IO 2 and IO 3 greatly outperform the remaining portfolios and avoid the Sharpe decreases of their unmanaged counterparts. The IO 1 avoids a couple of the crashes experienced by IO 1, but still earns very low returns. Panels A through C of Table 2 present performance results based on Eq. (3) of the volatilitymanaged IV portfolios as well as a long-short portfolio (IV 1 IV 3 ). Panel A shows that over each of the IV i and IV 2 has a beta with respect to IV i of about 0.6. The low- and medium-lta IV 1 earn statistically significant alpha and economically large appraisal ratios and utility gains (46% for both factors). In contrast, the high-lta IV 3 earns effectively no alpha with respect to IV 3. The volatility-managed long-short (IV 1 IV 3 ) also earns significant and economically large 9

11 alpha with respect to IV 1 IV 3. It is important to note that observing volatility-timing benefits for a long-short factor is insufficient to conclude that volatility timing improves performance for the long leg more than the short leg. One reason is that the volatility of a factor is a function of the volatilities of the long and short legs as well as the covariances between them. We find similar patterns over subsamples for IV 1 as Moreira and Muir (2017b) find for the market factor. Alphas of IV 1 are the largest in the early and late samples ( and ) and insignificant in the middle sample ( ). The low gains from volatility management derive from low variation in volatility over IV 2 only earns significant alpha in the early sample and IV 3 does not earn significant alpha in any sample. The (IV 1 IV 3 ) earns significant alpha in every sample. Furthermore, the alphas remain unchanged controlling for additional factors (MKT, SMB, and HML) in addition to the unmanaged portfolios. Thus, the benefits of volatility timing seem to robustly decline with IV. Panel D presents results analogous to those of Panel A, but for IO portfolios. The main result is the same between both Panels. The (high-lta) IO 1 exhibits no benefit from volatility management. In contrast, IO 2 and IO 3 earn significant alphas and have economically significant appraisal ratios that result in large utility gains of 68% 175%. The volatility-managed (IO 3 IO 1 ) also earns statistically significant alpha and Panel E shows that alphas are effectively unchanged when including the Fama-French factors. Overall, the evidence from Table 2 shows that the benefits from volatility managing long-equity portfolios concentrate in low- and medium-lta stocks and are insignificant for high-lta stocks. Thus, these results reject our main hypothesis and leave the anomalous returns of volatility-managed portfolios unexplained by frictions or rational models. The IO results especially serve as evidence against the Moreira-Muir slow-trading explanation of these anomalous returns because limits to short selling contribute to slow trading. It is also worth noting that the results in Table 2 must be unrelated to size, which is negatively correlated with IV and positively correlated with IO. Figures 2 and 3 illustrate why the Table 2 results work. In these figures, for each IV or IO tercile, respectively, we sort months into quintiles based on that month s volatility of IV i or IO i. We then plot the volatility, average return, and average return divided by average variance (risk-return 10

12 trade-off) within those quintiles. Figure 2 presents results for IV -sorted portfolios. Panels A, B, and C short that volatility is persistent from month-to-month for each IV i. However, Panels D and E show that on average, the returns of IV 1 and IV 2 are at most weekly related to volatility. This necessarily implies a negative risk-return tradeoff, which is seen in Panels G and H, and justifies the benefits of volatility timing. In contrast, Panel F shows a positive risk-return tradeoff for IV 3, which prevents superior performance of IV 3. The IV 3 result is interesting because it is more consistent with rational and frictionless theories than the negative trade-off for IV 1 and IV 2, in spite of the high arbitrage frictions in IV 3. The takeaways from Figure 3 parallel those of Figure 2, though are noisier because of the smaller sample size. Each IO i exhibits persistence in volatility in Panels A through C. However, Panels D through I show the risk-return trade-off is flatter for IO 1 than IO 2 and IO 3. The trade-off is negative, if anything, for IO 2 and IO Long-short portfolios Next we examine the performance of volatility-managed long-short factors constructed within each IV i and IO i. Independently of IV and IO, we sort stocks each month into quintiles based on each characteristic (ME, BM, MOM, INV, and OP ) associated with the Fama and French (1993, 2015) and Carhart (1997) factors (MKT, SMB, HML, MOM, RMW, and CMA). Within each IV or IO quintile, we construct high-minus-low or low-minus-high long-short portfolios (denoted, for example, by BM 5 1 or ME 1 5 ) that are signed to be positive on average. For each characteristic X, we also construct low-minus-high-iv (high-minus-low-io) difference portfolios, denoted IV 1 3 (X 5 1 ) or IV 1 3 (X 1 5 ) (IO 1 3 (X 5 1 ) or IO 1 3 (X 1 5 )). Table 3 presents CAPM alphas of the 3x5 and long-short portfolios constructed for each characteristic. Panels A through C show that in the sample, the spread in abnormal returns associated with ME, BM, and MOM increases with IV. This increase is large and significant for BM and MOM. Moreover, examining the subsample results shows that the spread in these three anomalies abnormal returns increases over time and are all significant in the most recent sample. Panels D and E show that over the sample, the abnormal returns associated with the 11

13 INV and OP anomalies typically increase with IV, although the significance of the increase is only marginally significant. Overall, anomalies appear to grow stronger with IV, consistent with its role as a limit to arbitrage. Panels F through J show a similar pattern for IO as for IV. Anomaly returns, especially the short legs, increase going from IO 3 to IO 1. This increase is significant for BM, MOM, and OP. These results are consistent with the limits-to-arbitrage property of IO. Table 4 presents alphas and utility gains from Eq. (3) for each long-short anomaly factor in Table 3. Results using IV include those for the different subsamples. Overall, the main takeaway is that no consistent pattern exists between the IV or IO rank and the performance of the managed factors, contrary to our hypothesis. Conditioning on IV or IO also frequently results in no performance gains of volatility management Mean-variance efficient portfolios Next, we apply the volatility-timing strategy to mean-variance-efficient (MVE) portfolios, which are constructed to have the maximum possible Sharpe ratios attainable from a set of factors. The alpha and utility gains of managed MVE portfolios approximate the potential gains of volatility-timing for investors who have access to many assets In-sample MVE portfolios For each IV i (IO i ), we estimate the unconditional in-sample (ex-post) MVE portfolio, denoted MV E IVi (MV E IOi ), constructed from one of two sets of factors. The first set of factors, denoted FF3+MOM, consists of the excess return on IV i (IO i ) as well as the long-short ME 1 5, BM 5 1, and MOM 5 1 factors constructed within IV i (IO i ) from Table 3. The second set of factors, denoted FF5+MOM, adds the corresponding OP 5 1 and INV 1 5. Panel A of Table 5 reports CAPM alphas for the unmanaged MV E IVi. Both the FF3+MOM and FF5+MOM MV E IV i alphas increase significantly and monotonically from IV 1 to IV 3. This result follows from the higher (hypothetical) returns to anomalies in high-lta stocks. Panel B reports performance statistics for the managed MV E IV i. Using the FF3+MOM factors, the MV E IV i each earn significant alpha with respect to MV E IVi over The highest 12

14 utility gain of 55% belongs to the low-lta MV IV1, although the MV E IV3 still has an economically large gain of 43%. Over , all three FF5+MOM MV E IV i also earn statistically significant alpha. Moreover, while, the economic significance of these alphas is lower than those of the FF3+MOM MV EIV i ranging from 9% to 26% it is still lowest in the high-lta IV 3 and lowest in IV 1. Panel C presents CAPM alphas of the unmanaged FF3+MOM and FF5+MOM MV E IOi. Similar to Panel A, The CAPM alphas of the MV E IOi increase significantly and monotonically going from low-lta to high-lta. This pattern reflects the greater returns to anomalies, especially the short legs, when LTA are high as indicated by low IO. Panel D shows the MV E IO i typically earn statistically significant alpha with respect to the unmanaged MV E IOi. However, the economic significance is two to three times higher for low-lta MV EIO 3 than the MV EIO 1. For the FF3+MOM and FF5+MOM MV EIO i, respectively, the utility gains of MV EIO3 are 59% and 63% compared to 18% and 31% for the MV E IO 1. Even if investors knew the in-sample MV E weights ex ante, they would have difficulty and bear large expenses to execute the strategies with high-lta stocks. For example, the success of the M V E depends critically on executing the short positions associated with each anomaly. In the low-io tercile, investors would likely find it prohibitively costly, if not impossible, to execute these short positions. Moreover, Novy-Marx and Velikov (2016) find that IV explains the crosssectional variation in stock-level transaction costs with an R 2 of 55%. Thus, the performance of the high-iv portfolios above is also overstated relative to what investors could realize. Conversely, the relatively high economic gains of volatility timing MV E IO3 and MV E IV1 are much more likely to be realizable because they have lower transaction costs as well as easier and less-costly shortselling. Thus, after frictions are considered, the economic gains to volatility managing in-sample M V E portfolios definitively appear to be greatest in low-lta stocks Out-of-sample MVE portfolios In-sample M V E portfolios overstate the maximum Sharpe ratios investors could obtain because their weights depend on future information. As a result, Moreira and Muir (2017b) note that the 13

15 gains from volatility managing in-sample M V E portfolios understate the true potential benefits of volatility timing. Hence, we estimate the benefits of volatility timing out-of-sample M V E portfolios. Let F IV it (Fit IO ) denote either the FF3+MOM or FF5+MOM factors for IV i (IO i ). For each month t > 120, we construct out-of-sample MVE portfolios, MV E IV i,t = b t 1 F IV t (MV E IOi = b t 1 F t IO ) by estimating b t 1 such that: b t = arg max b where SR(b) t 1 is the Sharpe ratio of the portfolio b F IV τ SR(b) t 1, (6) (b F IO τ ) over the window τ = 1,..., t 1. DeMiguel et al. (2009) show that out-of-sample estimates of tangency portfolios do not reliably outperform simple 1/N strategies that equal weight each asset in optimizations such as Eq. (6). Hence, we also apply our analysis to 1/N strategies constructed from the same factors as the MVE portfolios. We denote the latter (1/N) IVi or (1/N) IOi. Table 6 presents CAPM alphas of the unmanaged out-of-sample MVE portfolios and performance results of the volatility-managed counterparts. To take advantage of the maximum possible sample and thoroughly analyze subsamples while avoiding a profusion of panels, we relegate discussion of the FF5+MOM results, which are similar, to the Online Appendix. Each Panel corresponds to a choice of out-of-sample window. The estimation of the MVE portfolios begins with data 120 months (10 years) before the start of the window. For example, Panel A presents results over 1936:2 2015:12. The first observation (1936:2) of the MVE portfolios in Panel A is based on portfolio weights estimated over the prior 120 months (1927:2 1936:1). The second observation is based on the prior 121 months (1927:2 1936:2), and so on. Panel A presents CAPM alphas of the un-managed MV E IVi portfolios over Like their in-sample counterparts in Table 5, the CAPM alphas of these portfolios increase significantly going from low-to-high IV. Panel B presents Sharpe ratios of the unmanaged MV E IVi and performance results of the volatility-managed MV E IV i over The Sharpe ratios of the unmanaged portfolios are economically large, ranging from 0.91 to 1.22, which helps validate the out-of-sample application of our estimated MVE weights. For comparison, the market Sharpe ratio was 0.49 over the same 14

16 time period. The Sharpe ratios of the (1/N) portfolios are only slightly smaller, ranging from 0.78 to These high Sharpe ratios validate the use of the MV E and (1/N) portfolios as reasonable approximations to the mean-variance frontier in their respective groups of stocks. The alphas of the MV EIV i during this sample are all significant, however the utility gains for MV EIV 1 are more than three times as high as those of MV EIV 3 (65% vs 19%). The (1/N) IV 3 earns an insignificant alpha and has a lower Sharpe ratio than the unmanaged (1/N) IV3. In contrast, the (1/N) IV 1 earns a significant alpha and improves utility by 49%. Panel C shows that over the 1937:2 1955:12 subsample, none of the MV E IVi or (1/N) IOi earn significant alpha, perhaps because of the relatively short sample window. Panel D shows that in contrast to the results in the rest of the paper, two of the MV E IVi portfolios but none of the (1/N) IVi portfolios earn significant alpha during the sample that tends to exhibit weak gains of volatility management. However, the corresponding economic significance of these alphas is small (utility gains range from 0% to 12%). Panel D shows that over the recent subsample , the gains to volatility managing the MVE portfolios are generally more significant than those of the earlier samples. The MV EIV 3 and (1/N) IV3 do not earn significant alpha or generate meaningful utility gains. Volatility management even dramatically lowers the Sharpe ratios of MV EIV 3 and (1/N) IV3, from 1.13 to 0.84 and 1.04 to 0.54, respectively. In contrast, the MV EIV 1 and MV EIV 2 earn significant alphas and generate utility gains of 198% and 44%, respectively. The economic benefits of volatility management are also large for (1/N) IV1 and (1/N) IV2 during this time period with dramatic increases in Sharpe ratios and large utility gains of 200% and 27%, respectively. Panel F present CAPM alphas of unmanaged MV E IOi and (1/N) IOi. Like the MV E IVi alphas in Panel A, the alphas of MV E IOi increase significantly going from low-to-high LTA. However, the corresponding increase in alphas is insignificant for the (1/N) IOi portfolios. Panel G presents performance results for the MV E IO i and (1/N) IO i over 1996:2-2015:12. Both sets of portfolios earn significant alpha. However, the economic significance of the volatilitymanagement benefits increases dramatically going from low to high LTA. The utility gain of MV E IO 3 is 233% relative to the gain of 64% earned by MV E IO 1. Similarly, the utility gain 15

17 of (1/N) IO 3 (45%) is more than twice the utility gains of (1/N) IO 1 (27%). The utility gain of MV E IO1 may seem economically significant, however, it is again important to note that investing in MV E IO1 requires implementing the short legs of the constituent anomaly factors. This would almost certainly be prohibitively expensive or even impossible given the low IO. Overall, the results in Table 6 provide strong evidence that the ability of investors to improve the mean-variance efficiency of their portfolios is concentrated in stocks with the lowest LTA. 5. Conclusion Prior studies find that volatility managing portfolios scaling up when risk is low and down when risk is high produces significant alpha and utility gains. This phenomenon is not explained by rational asset pricing models, which would not be surprising if the phenomenon could instead be explained by arbitrage frictions. To the contrary, the results in this paper show that the economic gains from volatility management are actually concentrated in stocks with the lowest limits to arbitrage. Our empirical findings thus provide a strong challenge to rational and friction-driven models of asset pricing to be explained by future models. 16

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22 Cumulative log return IV1 unmanaged IV2 unmanaged IV3 unmanaged IV1 vol-managed IV2 vol-managed IV3 vol-managed (A) IV terciles Cumulative log return IO1 unmanaged IO2 unmanaged IO3 unmanaged IO1 vol-managed IO2 vol-managed IO3 vol-managed (B) IO terciles Figure 1: Cumulative log returns on unmanged and volatility-manged institutional ownership- and idiosyncratic volatility-tercile portfolios. Panel A (B) plots the cumulative returns to a buy-and-hold strategy versus a volatility-managed strategy for each idiosyncratic volatility (IV)-tercile (institutional ownership (IO)-tercile) portfolio from 1926 to 2015 (1986 to 2015). The y-axis is on a log scale and the volatility-managed strategies have the same unconditional monthly standard deviation as their unmanaged counterparts. 21

23 Volatility (%) Volatility (%) Volatility (%) (A) Volatility, IV1 (B) Volatility, IV2 (C) Volatility, IV3 Avg return (%) Avg return (%) Avg return (%) (D) Avg return, IV1 (E) Avg return, IV2 (F) Avg return, IV3 E(R)/Var(R) E(R)/Var(R) E(R)/Var(R) (G) E(r)/ 2 (r), IV1 (H) E(r)/ 2 (r), IV2 (I) E(r)/ 2 (r), IV3 Figure 2: Sorts on previous month s volatility quintile by idiosyncratic-volatility. For each IV-tercile portfolio, we use the monthly time series of realized volatility to sort the following months returns into five buckets. The lowest, 1 looks at the properties of returns over the month following the lowest 20% of realized volatility months. We show the average return over the next month, the standard deviation over the next month, and the average return divided by variance for each tercile. 22

24 Volatility (%) Volatility (%) Volatility (%) (A) Volatility, IO1 (B) Volatility, IO2 (C) Volatility, IO3 Avg return (%) Avg return (%) Avg return (%) (D) Avg return, IO1 (E) Avg return, IO2 (F) Avg return, IO3 E(R)/Var(R) E(R)/Var(R) E(R)/Var(R) (G) E(r)/ 2 (r), IO1 (H) E(r)/ 2 (r), IO2 (I) E(r)/ 2 (r), IO3 Figure 3: Sorts on previous month s volatility quintile by institutional ownership. For each IO-tercile portfolio, we use the monthly time series of realized volatility to sort the following months returns into five buckets. The lowest, 1 looks at the properties of returns over the month following the lowest 20% of realized volatility months. We show the average return over the next month, the standard deviation over the next month, and the average return divided by variance for each tercile. 23

25 Table 1: Average returns and CAPM alphas of unmanaged idiosyncratic volatility (IV)- and institutional ownership (IO)-tercile portfolios Panels A and B, respectively, present average excess returns and CAPM alphas for each of the idiosyncratic volatility (IV) portfolios IV 1, IV 2, IV 3 or the low-minus-high factor IV 1 IV 3. Panels C and D present the same statistics for each of the institutional ownership (IO) portfolios (IO 1, IO 2, IO 3 ), or the high-minus-low-io factor (IO 3 IO 1 ). t-statistics are below point estimates in parentheses. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. R 2 denotes adjusted R 2 in all tables. In Panels A and B the sample period is 1926:8-2015:12 (N=1073). In Panels C and D the sample is 1986:1-2015:12 (N=360). Panel A: Average returns of IV portfolios IV 1 IV 2 IV 3 IV 1 IV 3 r e 8.32*** 9.32*** ** (4.55) (3.52) (1.12) (2.25) Panel B: CAPM alphas of IV portfolios IV 1 IV 2 IV 3 IV 1 IV 3 β 0.92*** 1.30*** 1.45*** -0.53*** (148.98) (56.25) (33.12) (-11.09) α 1.23*** *** 8.78*** (4.30) (-1.23) (-4.85) (4.91) R Panel C: Average returns of IO portfolios IO 1 IO 2 IO 3 IO 3 IO 1 r e * 7.96*** 9.98*** (-0.58) (1.80) (2.67) (4.21) Panel D: CAPM alphas of IO portfolios IO 1 IO 2 IO 3 IO 3 IO 1 β 0.96*** 0.98*** 1.04*** 0.08 (22.65) (42.10) (74.55) (1.60) α -9.35*** -2.15** *** (-4.24) (-2.12) (0.07) (3.89) R